Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. y''-7y'+6y =e^t + delta(t-3) + delta(t-5) y(0) = 0 y'(0) = 0

Albarellak 2021-01-19 Answered
Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions.
y7y+6y=et+δ(t3)+δ(t5)
y(0)=0
y(0)=0
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Expert Answer

Corben Pittman
Answered 2021-01-20 Author has 83 answers
Step 1
Here we use table of Laplace transforms and some properties of Laplace transform.
Step 2
y7y+6y=et+δ(t3)+δ(t5)
y(0)=0
y(0)=0
Apply laplace transformL{y"}7L{y}+6L{y}=L{et}+L{δ(t3)}+L{δ(t5)}
use L{δ(tc)}=ecs
(s2Y(s)sY(0)y(0))7(sY(s)y(0))+6y(s)=1(s1)+e3s+e5s
(s27s+6)Y(s)=1(s1)+e3s+e5s
Y(s)=1(s1)(s27s+6)+e3s(s27s+6)+e5s(s27s+6)
Now apply inverse laplace transform
L1{Y(s)}=L1{1(s1)(s27s+6)}+L1{e3s(s27s+6)}+L1{e5s(s27s+6)}
Now use paricl and use table of Laplace transforms
y(t)=16tet16e(t3)δ(t3)16et5δ(t5)
here we use properties L1{F(sc)}=ectf(t)
L1{ecsF(s)}=y(t)f(t)
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